Finite elements for elliptic problems with stochastic coefficients

Frauenfelder, Philipp; Schwab, Christoph; Todor, Radu Alexandru

doi:10.1016/j.cma.2004.04.008

Cited by 341 publications

(345 citation statements)

References 7 publications

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“…A typical approach is to conduct a Galerkin projection to minimize the error of the finite-order gPC expansion, and the resulting set of equations for the expansion coefficients are deterministic and can be solved via conventional numerical techniques. This has been done in various applications and proved to be very effective (cf, [1,7,11,17,21,39,40,38]). …”

Section: Stochastic Modellingmentioning

confidence: 99%

Parametric uncertainty analysis of pulse wave propagation in a model of a human arterial network

Xiu

Sherwin

2007

Journal of Computational Physics

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Reduced models of human arterial networks are an efficient approach to analyse quantitative macroscopic features of human arterial flows. The justification for such models typically arise due to the significantly long wavelength associated with the system in comparison to the lengths of arteries in the networks. Although these types of models have been employed extensively and many issues associated with their implementations have been widely researched, the issue of data uncertainty has received comparatively little attention. Similar to many biological systems, a large amount of uncertainty exists in the value of the parameters associated with the models. Clearly reliable assessment of the system behaviour can not be made unless the effect of such data uncertainty is quantified.In this paper we present a study of parametric data uncertainty in reduced modelling of human arterial networks which is governed by a hyperbolic system. The uncertain parameters are modelled as random variables and the governing equations for the arterial network therefore become stochastic. This type stochastic hyperbolic systems have not been previously systematically studied due to the difficulties introduced by the uncertainty such as a potential change in the mathematical character of the system and imposing boundary conditions. We demonstrate how the application of a high-order stochastic collocation method based on the generalized polynomial chaos expansion, combined with a discontinuous Galerkin spectral/hp element discretisation in physical space, can successfully simulate this type of hyperbolic system subject to restrictions on the uncertainty interval. Building upon a numerical study of propagation of uncertainty and sensitivity in a simplified model with a single bifurcation, a systematical parameter sensitivity analysis is conducted on the wave dynamics in a multiple bifurcating human arterial network. Using the physical understanding of the dynamics of pulse waves in these types of networks we are able to provide an insight into the results of the stochastic simulations, thereby demonstrating the effects of uncertainty in physiologically accurate human arterial networks.

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Section: Stochastic Modellingmentioning

confidence: 99%

Parametric uncertainty analysis of pulse wave propagation in a model of a human arterial network

Xiu

Sherwin

2007

Journal of Computational Physics

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“…The well-posedness of (2.1)-(2.2), primal variational formulations, and approximation schemes based on finite element spatial discretizations have been widely studied (see [9], [4], [5], [3], [13], [23], [35]). Stochastic Galerkin approximation, specifically, has been studied in [4] and [9] and solvers for the resulting symmetric positive definite linear systems have been studied in [27], [32] and [30].…”

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confidence: 99%

Preconditioning Stochastic Galerkin Saddle Point Systems

Powell¹,

Ullmann²

2010

SIAM J. Matrix Anal. & Appl.

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Abstract. Mixed finite element discretizations of deterministic second-order elliptic partial differential equations (PDEs) lead to saddle point systems for which the study of iterative solvers and preconditioners is mature. Galerkin approximation of solutions of stochastic second-order elliptic PDEs, which couple standard mixed finite element discretizations in physical space with global polynomial approximation on a probability space, also give rise to linear systems with familiar saddle point structure. For stochastically nonlinear problems, the solution of such systems presents a serious computational challenge. The blocks are sums of Kronecker products of pairs of matrices associated with two distinct discretizations and the systems are large, reflecting the curse of dimensionality inherent in most stochastic approximation schemes. Moreover, for the problems considered herein, the leading blocks of the saddle point matrices are block-dense and the cost of a matrix vector product is non-trivial.We implement a stochastic Galerkin discretization for the steady-state diffusion problem written as a mixed firstorder system. The diffusion coefficient is assumed to be a lognormal random field, approximated via a nonlinear function of a finite number of unbounded random parameters. We study the resulting saddle point systems and investigate the efficiency of block-diagonal preconditioners of Schur complement and augmented type, for use with minres. By introducing so-called Kronecker product preconditioners we improve the robustness of cheap, mean-based preconditioners with respect to the statistical properties of the stochastically nonlinear diffusion coefficients.

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“…PC methods seem attractive because the underlying polynomial expansions can converge quickly [5,21]. These methods can also be used to produce easily conditionalized polynomial models [23].…”

Section: Introductionmentioning

confidence: 99%

“…This decoupling is accomplished by solving a generalized eigenvalue problem. Using this decoupling, Frauenfelder et al [21] studied an elliptic equation, with conductivity linear in each of the underlying random variates. The DOV method has also been applied to a linear transport model [43].…”

Section: Introductionmentioning

confidence: 99%

An analysis of polynomial chaos approximations for modeling single-fluid-phase flow in porous medium systems

Rupert

Miller

2007

Journal of Computational Physics

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We examine a variety of polynomial-chaos-motivated approximations to a stochastic form of a steady state groundwater flow model. We consider approaches for truncating the infinite dimensional problem and producing decoupled systems. We discuss conditions under which such decoupling is possible and show that to generalize the known decoupling by numerical cubature, it would be necessary to find new multivariate cubature rules. Finally, we use the acceleration of Monte Carlo to compare the quality of polynomial models obtained for all approaches and find that in general the methods considered are more efficient than Monte Carlo for the relatively small domains considered in this work. A curse of dimensionality in the series expansion of the log-normal stochastic random field used to represent hydraulic conductivity provides a significant impediment to efficient approximations for large domains for all methods considered in this work, other than the Monte Carlo method.

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Finite elements for elliptic problems with stochastic coefficients

Cited by 341 publications

References 7 publications

Parametric uncertainty analysis of pulse wave propagation in a model of a human arterial network

Parametric uncertainty analysis of pulse wave propagation in a model of a human arterial network

Preconditioning Stochastic Galerkin Saddle Point Systems

An analysis of polynomial chaos approximations for modeling single-fluid-phase flow in porous medium systems

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